Theorem csbnestg 2007

Description: Nest the composition of two substitutions.

Assertion

Ref	Expression
csbnestg	|- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)

Distinct variable groups:   x,C   x,y

Proof of Theorem csbnestg

Step	Hyp	Ref	Expression
1		csbcog 1978	. . . . 5 |- (A e. V -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / z]_[_z / y]_C)
2	1	adantr 389	. . . 4 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / z]_[_z / y]_C)
3		visset 1788	. . . . . . . 8 |- w e. V
4		csbnestglem 2006	. . . . . . . 8 |- ((w e. V /\ A.x B e. V) -> [_w / x]_[_B / z]_[_z / y]_C = [_[_w / x]_B / z]_[_z / y]_C)
5	3, 4	mpan 692	. . . . . . 7 |- (A.x B e. V -> [_w / x]_[_B / z]_[_z / y]_C = [_[_w / x]_B / z]_[_z / y]_C)
6	5	csbeq2dv 1990	. . . . . 6 |- ((A.x B e. V /\ A e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / w]_[_[_w / x]_B / z]_[_z / y]_C)
7	6	ancoms 436	. . . . 5 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / w]_[_[_w / x]_B / z]_[_z / y]_C)
8		csbnestglem 2006	. . . . . 6 |- ((A e. V /\ A.w[_w / x]_B e. V) -> [_A / w]_[_[_w / x]_B / z]_[_z / y]_C = [_[_A / w]_[_w / x]_B / z]_[_z / y]_C)
9		csbexg 1979	. . . . . . . 8 |- ((w e. V /\ A.x B e. V) -> [_w / x]_B e. V)
10	3, 9	mpan 692	. . . . . . 7 |- (A.x B e. V -> [_w / x]_B e. V)
11	10	19.21aiv 1268	. . . . . 6 |- (A.x B e. V -> A.w[_w / x]_B e. V)
12	8, 11	sylan2 451	. . . . 5 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_[_w / x]_B / z]_[_z / y]_C = [_[_A / w]_[_w / x]_B / z]_[_z / y]_C)
13		csbcog 1978	. . . . . . 7 |- (A e. V -> [_A / w]_[_w / x]_B = [_A / x]_B)
14	13	csbeq1d 1975	. . . . . 6 |- (A e. V -> [_[_A / w]_[_w / x]_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
15	14	adantr 389	. . . . 5 |- ((A e. V /\ A.x B e. V) -> [_[_A / w]_[_w / x]_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
16	7, 12, 15	3eqtrd 1487	. . . 4 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
17	2, 16	eqtr3d 1485	. . 3 |- ((A e. V /\ A.x B e. V) -> [_A / x]_[_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
18		hba1 979	. . . . 5 |- (A.x B e. V -> A.xA.x B e. V)
19		csbcog 1978	. . . . . 6 |- (B e. V -> [_B / z]_[_z / y]_C = [_B / y]_C)
20	19	a4s 960	. . . . 5 |- (A.x B e. V -> [_B / z]_[_z / y]_C = [_B / y]_C)
21	18, 20	csbeq2d 1989	. . . 4 |- ((A.x B e. V /\ A e. V) -> [_A / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / y]_C)
22	21	ancoms 436	. . 3 |- ((A e. V /\ A.x B e. V) -> [_A / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / y]_C)
23		csbexg 1979	. . . 4 |- ((A e. V /\ A.x B e. V) -> [_A / x]_B e. V)
24		csbcog 1978	. . . 4 |- ([_A / x]_B e. V -> [_[_A / x]_B / z]_[_z / y]_C = [_[_A / x]_B / y]_C)
25	23, 24	syl 10	. . 3 |- ((A e. V /\ A.x B e. V) -> [_[_A / x]_B / z]_[_z / y]_C = [_[_A / x]_B / y]_C)
26	17, 22, 25	3eqtr3d 1491	. 2 |- ((A e. V /\ A.x B e. V) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)
27		elisset 1792	. 2 |- (A e. R -> A e. V)
28		elisset 1792	. . 3 |- (B e. S -> B e. V)
29	28	19.20i 968	. 2 |- (A.x B e. S -> A.x B e. V)
30	26, 27, 29	syl2an 454	1 |- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950   = wceq 1099   e. wcel 1105  Vcvv 1786  [_csb 1972

This theorem is referenced by:  sbcnestg 2009  csbco3g 2011

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913  df-csb 1973

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